Question

Write the equation of a polynomial function P(x) that is degree 3 or higher and satisfies the following conditions:

The function has at least two intervals where P(x) < 0.
The function has an IRoC = 0 at one of the points on the x-axis.
P(3) = 0.
P(-5) = 6.
P(x) has at most 5 turning points.
The function must have a LOCAL MINIMUM AND LOCAL MAXIMUM point.
b) Graphically prove that your function satisfies all five conditions above.

Answer 1

The polynomial function that satisfies the given conditions is
`\( P(x) = (x - 3)(x + 5)^2 \).`

Step-by-step explanation:

To fulfill the conditions, we start with the roots at
`\( x = 3 \)` (from
`\( P(3) = 0 \))` and
`\( x = -5 \)` (from
`\( P(-5) = 6 \))`. These roots ensure the function crosses the x-axis at these points. To introduce at least two intervals where
`\( P(x) < 0 \)`, we use
`\( (x + 5)^2 \)` to make sure the interval
`\( x < -5 \)` has a negative output. The factor
`\( (x - 3) \)` guarantees that the interval
`\( x > 3 \)` also produces negative values.

Now, for the Inflection Point (IP) with
`\( IRoC = 0 \)`, the double root at
`\( x = -5 \)` achieves this. The factor
`\( (x + 5)^2 \)` ensures that the first derivative at
`\( x = -5 \)` is zero, making it an Inflection Point with a zero Instantaneous Rate of Change.

To limit turning points to at most 5, we choose a cubic polynomial with two real roots. This also ensures there is at least one turning point. Finally, to introduce a Local Minimum and Local Maximum, we use the fact that a cubic polynomial has one point of inflection, giving us the desired shape.

In graphical verification, plotting
`\( P(x) \)` reveals the roots at
`\( x = 3 \)` and
`\( x = -5 \)`, satisfying the conditions. The double root at
`\( x = -5 \)` contributes to the Inflection Point with
`\( IRoC = 0 \)`, and the overall shape adheres to the limits on turning points, showcasing a Local Minimum and Local Maximum as well.

Answer 2

The cubic polynomial P(x) = (-3/196)(x - 3)(x + 2)^2 satisfies all the conditions by having the right intercepts, intervals of negativity, a repeated root causing IRoC to be zero, and having at most 2 turning points for local extremas.

Step-by-step explanation:

To write the equation of a polynomial function P(x) that satisfies the given conditions, let's construct a cubic function, as this is the simplest polynomial that can fulfill the requirements.

One such function could be P(x) = a(x - 3)(x + 2)^2, where a determines the wideness of the graph and the direction it opens.

Since we need at least two intervals where P(x) < 0 and a point on the x-axis where the Instantaneous Rate of Change (IRoC) = 0, we can choose a point on the x-axis where the polynomial has a repeated root.

Let's choose a to be positive because we want the graph to open upwards.

We know that P(3) = 0, which suggests that (x - 3) is a factor of our polynomial.

P(-5) = 6 indicates that when x = -5, the y-value is positive, which helps us determine a possible placement of a local minimum.

To ensure that P(x) has at most 5 turning points, we can stick with a cubic function, which by nature has at most 2 turning points.

Therefore, our polynomial will have a local minimum and local maximum.

Using these conditions, we may choose the function P(x) = (x - 3)(x + 2)^2.

However, to satisfy the condition P(-5) = 6, we'll need to find the correct coefficient.

Substituting x = -5, we get -8(a)(49) = 6, which gives us a = -6/392. The final function is P(x) = (-3/196)(x - 3)(x + 2)^2.

To graphically prove that the function satisfies all the conditions, we need to create a graph.

This graph will show:

Two intervals where P(x) < 0.

A point on the x-axis where the derivative is 0 (at x = -2).

Intercepts at x = 3 and x = -2 (with a double root at -2 causing the derivative to be 0).

The point (-5, 6) will be on the graph.

A local minimum and local maximum.

The graph of this polynomial will clearly illustrate points of intercept with the x-axis, ascends and descends denoting local minima and maxima, and regions where the function dips below the x-axis.